We work over $\mathbb{C}$. Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra (the question is still relevant for the non-affine case, but I'm specifically interested in the affine case). Suppose that $V$ is an $\textit{integrable}$ representation of $\mathfrak{g}$ with the same character as the adjoint representation of $\mathfrak{g}$. Then must it be true that $V\cong \mathfrak{g}$?
If not, is a minimal (or at least a small) presentation of the adjoint representation known, so that one can check if $V\cong\mathfrak{g}$?
